Maximum bisections of graphs without cycles of length 4

Abstract

A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most 1, and its size is the number of edges which go across the two parts. Let Ck be a cycle of length k, and let G be a C4-free graph with n vertices, m edges and vertex degrees d1,…,dn. Lin and Zeng proved that if G does not contain C6 and has a perfect matching, then G admits a bisection of size at least m/2+Ω(∑i=1ndi). This extends a celebrated bound given by Shearer on Max-Cut of triangle-free graphs. In this paper, we establish a similar result by replacing C6 with θ(1,2,4), θ(2,3,3) and θ(3,3,3), where θ(ℓ1,ℓ2,ℓ3) denotes the graph consisting of three internally disjoint paths of length ℓ1, ℓ2 and ℓ3, respectively, each with the same endpoints. We also note that the bound is tight for certain polarity graphs.